Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Computable and partial computable functions
Definition
Fix effective encodings of sets and .
A function is partial computable when some deterministic Turing machine computes it in the sense of Partial functions computed by a machine under fixed encodings.
If , so the machine halts with the correct output on every input in , then is computable or total computable.
Remarks
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"Computable" is the total case of "partial computable"; the distinction is whether divergence is allowed outside a proper domain.
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The encoded machine works on binary words, but the represented function lives on the underlying sets and through the chosen encodings.
Depends on
Used by
- Decidable and recognizable languages Definition
Dependency tree · two levels
4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Jean Gallier and Jocelyn Quaintance, Introduction to the Theory of Computation: Some Notes for CIS511 (standard reference, not scraped)
- John Watrous, Introduction to the Theory of Computing, Lecture 18: Further discussion of computability (standard reference, not scraped)